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ORACLE-EFFICIENT NONPARAMETRIC ESTIMATION OF AN ADDITIVE MODEL WITH AN UNKNOWN LINK FUNCTION

Published online by Cambridge University Press:  11 October 2010

Joel L. Horowitz*
Affiliation:
Northwestern University
Enno Mammen
Affiliation:
University of Mannheim
*
*Address Correspondence to Joel L. Harowitz, Department of Economics, Northwestern University, Evanston, IL 60208-2600 U.S.A.; E-mail: joel-horowitz@northwestern.edu.

Abstract

This paper describes an estimator of the additive components of a nonparametric additive model with an unknown link function. When the additive components and link function are twice differentiable with sufficiently smooth second derivatives, the estimator is asymptotically normally distributed with a rate of convergence in probability of n−2/5. This is true regardless of the (finite) dimension of the explanatory variable. Thus, the estimator has no curse of dimensionality. Moreover, the asymptotic distribution of the estimator of each additive component is the same as it would be if the link function and the other components were known with certainty. Thus, asymptotically there is no penalty for not knowing the link function or the other components.

Type
ARTICLES
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Breiman, L. & Friedman, J.H. (1985) Estimating optimal transformations for multiple regression and correlation. Journal of the American Statistical Association 80, 580598.CrossRefGoogle Scholar
Buja, A., Hastie, T., & Tibshirani, R.J. (1989) Linear smoothers and additive models. Annals of Statistics 17, 453510.Google Scholar
Carrasco, M., Florens, J.-P., & Renault, E. (2007) Linear inverse problems in structural econometrics: Estimation based on spectral decomposition and regularization. In Leamer, E.E. & Heckman, J.J. (eds.), Handbook of Econometrics, vol. 6, pp. 56335746. North-Holland.CrossRefGoogle Scholar
Chen, R., Härdle, W., Linton, O.B., & Severance-Lossin, E. (1996) Estimation in additive nonparametric regression. In Härdle, W. & Schimek, M. (eds.), Proceedings of the COMPSTAT Conference Semmering. Phyisika Varlag.Google Scholar
Fan, J., Härdle, W., & Mammen, E. (1998) Direct estimation of low-dimensional components in additive models. Annals of Statistics 26, 943971.CrossRefGoogle Scholar
Hastie, T.J. & Tibshirani, R.J. (1990) Generalized Additive Models. Chapman & Hall.Google Scholar
Horowitz, J.L. (2001) Nonparametric estimation of a generalized additive model with an unknown link function. Econometrica 69, 499513.CrossRefGoogle Scholar
Horowitz, J.L., Klemelä, J., & Mammen, E. (2006) Optimal estimation in additive regression models. Bernoulli 12, 271298.CrossRefGoogle Scholar
Horowitz, J.L. & Mammen, E. (2004) Nonparametric estimation of an additive model with a link function. Annals of Statistics 32, 24122443.CrossRefGoogle Scholar
Horowitz, J.L. & Mammen, E. (2007) Rate-optimal estimation for a general class of nonparametric regression models with unknown link functions. Annals of Statistics 35, 25892619.CrossRefGoogle Scholar
Hristache, M., Juditsky, A., & Spokoiny, V. (2001) Structure adaptive approach for dimension reduction. Annals of Statistics 29, 132.CrossRefGoogle Scholar
Ichimura, H (1993) Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. Journal of Econometrics 58, 71120.CrossRefGoogle Scholar
Jennrich, R.I. (1969). Asymptotic properties of non-linear least squares estimators. Annals of Mathematical Statistics 40, 633643.CrossRefGoogle Scholar
Juditsky, A.B., Lepski, O.V., & Tsybakov, A.B. (2009) Nonparametric estimation of composite functions. Annals of Statistics 37, 13601404.CrossRefGoogle Scholar
Linton, O.B. (2000) Efficient estimation of generalized additive nonparametric regression models. Econometric Theory 16, 502523.CrossRefGoogle Scholar
Linton, O.B. & Härdle, W. (1996) Estimating additive regression with known links. Biometrika 83, 529540.CrossRefGoogle Scholar
Linton, O.B. & Nielsen, J.P. (1995) A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika 82, 93100.CrossRefGoogle Scholar
Mammen, E., Linton, O.B., & Nielsen, J.P. (1999) The existence and asymptotic properties of backfitting projection algorithm under weak conditions. Annals of Statistics 27, 14431490.CrossRefGoogle Scholar
Newey, W.K. (1994) Kernel estimation of partial means and a general variance estimator. Econometric Theory 10, 233253.CrossRefGoogle Scholar
Newey, W.K. (1997) Convergence rates and asymptotic normality for series estimators. Journal of Econometrics 79, 147168.CrossRefGoogle Scholar
Opsomer, J.D. (2000) Asymptotic properties of backfitting estimators. Journal of Multivariate Analysis 73, 166179.CrossRefGoogle Scholar
Opsomer, J.D. & Ruppert, D. (1997) Fitting a bivariate additive model by local polynomial regression. Annals of Statistics 25, 186211.CrossRefGoogle Scholar
Serfling, R.J. (1980) Approximation Theorems of Mathematical Statistics. Wiley.CrossRefGoogle Scholar
Stone, C.J. (1985) Additive regression and other nonparametric models. Annals of Statistics 13, 689705.CrossRefGoogle Scholar
Stone, C.J. (1986) The dimensionality reduction principle for generalized additive models. Annals of Statistics 14, 590606.CrossRefGoogle Scholar
Stone, C.J. (1994) The use of polynomial splines and their tensor products in multivariate function estimation. Annals of Statistics 2, 118171.Google Scholar
Tjøstheim, D. & Auestad, B.H. (1994) Nonparametric identification of nonlinear time series: Projections. Journal of the American Statistical Association 89, 13981409.Google Scholar